## 7th and 8th Grade, December, 2019

C.1273. Four llamas are grazing. They can finish a meadow in 2 days if each one of them eats for 2 hours every day. How long would it take 2 llamas to finish the same meadow if each would eat for 8 hours every day?

C.1274. We grouped the positive whole numbers the following way:

(1); (2, 3); (4, 5, 6); (7, 8, 9, 10); (11, 12, 13, 14, 15); …

Which group contains the number 2010, and where in that group is it?

C.1275. 2163 is a four-digit number in which the digit in the units place is three times the digit in the hundreds place, and the digit in the tens place is twice the sum of the digits in the hundreds and the thousands place. Is it true that all of these kinds of numbers are divisible by 3?

C.1276. Replace the letters A, B, C, D, E, F and G, by the numbers 1, 2, 3, 4, 5, 6 and 7, so that the sums of the numbers of the quadrilaterals in the three corners are all 15. What numbers can replace the letter A?

C.1277. How many 3-digit numbers have all different digits written in either increasing or decreasing order?

C.1278. How many 4-digit numbers are there in which the sums of the first two and the last two digits are the same? (Such as in 4536, or 1506.)

C.1279. In a given year none of the following dates fall on the weekend (on Saturday or Sunday): August 20, September 1, October 23, December 26. What day of the week is the last day of that year?

C.1280. Take any given order of the numbers 1, 2, 3, 4, …, n. Create a new number sequence by subtracting 1, 2, 3, 4, …, and n from the elements of that order, respectively. Prove that if n is an odd number then the product of the elements of the new sequence is even.